Thermal Freeze-Out and the Abundance of Weakly Interacting Massive Particles

Weakly interacting massive particles, commonly abbreviated as WIMPs, remain a central pillar of particle cosmology. The freeze-out mechanism provides an elegant narrative that connects microscopic particle interactions to the macroscopic composition of the universe. In the hot early cosmos, particles and antiparticles were in thermal equilibrium with the primordial plasma, continuously annihilating and being created through inverse reactions. As the universe expanded and cooled, the equilibrium number density of a massive species dropped exponentially. Eventually, the expansion rate overwhelmed the annihilation rate, causing the species to fall out of chemical equilibrium. The comoving number density thereafter stayed nearly constant, yielding a relic abundance that we can observe today as dark matter. The calculator on this page encapsulates the essential physics of the freeze-out process in a simple interface, allowing users to estimate the present-day density parameter Ω_χh² from a candidate's mass and annihilation cross-section.

The freeze-out of a WIMP is governed by the Boltzmann equation for the number density n:

Here, H is the Hubble expansion rate, n_eq is the equilibrium number density, and ⟨σv⟩ denotes the thermally averaged annihilation cross-section times relative velocity. While solving this differential equation exactly requires numerical integration, analytic approximations capture the main dependence on particle physics parameters. We introduce the dimensionless quantity x = m_χ/T, where T is the temperature. Freeze-out occurs around x_f ≈ 20–30 for typical WIMPs. An iterative solution to the transcendental equation

$x{f}_{}=\text{ln}[\frac{0.038}{g}$

provides a useful estimate. In this expression, g represents the internal degrees of freedom of the WIMP, g_* counts the effective number of relativistic degrees of freedom at freeze-out, and M_Pl is the Planck mass. After freeze-out, the comoving abundance remains constant. The resulting present-day relic density can be written in terms of the annihilation cross-section as

${\Omega }_{\chi }{h}^2\approx \frac{0.1}{{⟨}_{\sigma }/3\times {10}^{-26}\text{cm}3/\text{s}}$

This approximate inverse relationship underscores the so-called WIMP miracle: a cross-section characteristic of weak interactions naturally yields a relic density of order unity when expressed in cosmological units. The calculator adopts this scaling as its baseline. Users supply a mass m_χ in GeV and the annihilation cross-section ⟨σv⟩ in cm³/s. Internally, the tool computes x_f using the iterative relation above (assuming g = 2 and g_* = 90 as typical values) and then evaluates Ω_χh². The resulting classification indicates whether the predicted relic density exceeds, matches, or falls short of the observed cold dark matter value of approximately 0.12.

To interpret the output, recall that cross-sections larger than 3×10⁻²⁶ cm³/s deplete the relic abundance, while smaller cross-sections leave too many WIMPs. In many models, the cross-section scales inversely with m_χ², but resonances, coannihilations, and threshold effects can complicate this picture. The calculator deliberately ignores these subtleties to remain general. Nevertheless, the derived x_f and freeze-out temperature T_f = m_χ/x_f offer a reasonable indication of when in cosmic history freeze-out occurred. For example, a 100 GeV WIMP with the canonical cross-section freezes out at T_f ≈ 5 GeV, corresponding to microsecond-old universe times.

Understanding the freeze-out calculation provides insight into how relic densities depend on fundamental constants. The table below illustrates sample outputs for different masses and cross-sections:

mχ (GeV)	⟨σv⟩ (cm³/s)	xf	Ωχh2	Classification
100	3×10−26	23	0.12	Matches
500	1×10−26	25	0.36	Overabundant

These examples show how increasing the mass at fixed cross-section raises x_f slightly but leaves the relic density predominantly controlled by ⟨σv⟩. In realistic model-building, coannihilations with nearby states or velocity-dependent cross-sections modify the simple scaling. Nevertheless, the freeze-out paradigm serves as an instructive guide for evaluating hypothetical dark matter candidates across a wide parameter space.

The freeze-out story connects to several deep theoretical and observational efforts. On the theoretical side, supersymmetry, extra-dimensional models, and theories with hidden-sector gauge interactions all provide WIMP candidates whose masses and couplings can be plugged into this calculator. From the observational perspective, direct detection experiments measure the scattering cross-section of dark matter with nuclei, while indirect detection searches look for annihilation products such as gamma rays or antimatter. If a signal is observed, comparing the inferred annihilation cross-section with the relic density estimate offers a consistency check on whether the candidate was thermally produced. Likewise, collider experiments like the LHC can produce missing-energy events that might correspond to WIMPs; the calculator helps assess whether such particles could constitute all of dark matter.

Beyond the standard cosmological history, modifications such as an early period of matter domination, scalar-tensor gravity, or additional relativistic species (altering g_*) can shift the freeze-out abundance. Entropy injection after freeze-out can dilute relics, while non-thermal production channels can augment them. The calculator assumes none of these exotic scenarios, but the underlying formulae can be adapted by substituting the appropriate g_* or expansion history. This flexibility has encouraged a vast literature exploring non-standard freeze-out mechanisms like co-scattering, cannibalism, and bound-state effects, each of which leaves distinct imprints on the relic density.

For students learning cosmology, working through the derivation of the freeze-out equation offers a concrete application of thermodynamics in an expanding universe. Starting from the Boltzmann equation, one changes variables to the comoving yield Y = n/s, where s is the entropy density, and finds an equation of the form

with λ ∝ M_Pl m_χ ⟨σv⟩. Solving this equation yields the relic abundance. The calculator implicitly implements a simplified version of this solution, illustrating how microscopic physics feeds into macroscopic observables. Such exercises deepen intuition about the interplay between particle properties and cosmic evolution.

In summary, the WIMP freeze-out relic density calculator provides a quick yet informative window into one of the most influential ideas in dark matter physics. By entering just a mass and annihilation cross-section, users can estimate the abundance and assess whether a candidate aligns with cosmological observations. While full model assessments require more detailed computations, this tool highlights the parametric dependencies that underlie the celebrated WIMP miracle and continues to inspire experimental searches across the globe.